Self-Avoiding-Walks-Final.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# How To Avoid Oneself: A Monte-Carlo Study"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#importing the libraries and modules we will need, run this first\n",
    "\n",
    "import numpy as np\n",
    "from numpy.random import randint\n",
    "import matplotlib.pyplot as plt\n",
    "import time\n",
    "from matplotlib.ticker import PercentFormatter\n",
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Drunkard's Walk: A Simple 2D Random Walk "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A **random walk** (or more evocatively, a drunkard's walk) is a common mathematical model and thought experiment that goes as follows: a man walks out of his favourite pub to go home, but with his navigation faculties significantly impaired. At every step he takes, he has a roughly equal probability of walking North, East, South or West. After a long enough time, where are we likely to find him, and what is the path he has followed? It turns out that in $2D$, the probability distribution for the straight line distanced covered by the man can be described by a Gaussian (normal) curve, specifically the Raleigh distribution,\n",
    "\\begin{equation}\n",
    "    P(r)=\\frac{2r}{N}e^{-r^2/N}\n",
    "\\end{equation}    \n",
    "\n",
    "and the distance ($R$) varies with respect to the number of steps ($N$) as\n",
    "\\begin{equation}\n",
    "    R^2 \\propto N\n",
    "\\end{equation}\n",
    "\n",
    "While the model seems laughably primitive, it was used by Albert Einstein to explain Brownian motion of a colloidal particle and by Louis Bachelier to model the stock prices of French government bonds around the early 20th century, and still serves as a very good statistical model for diffusion phenomena and for the Black-Scholes formula for options pricing (as two examples). Let us now see what a walk like that would look like by writing a simple program, starting from the origin $(0,0)$ and using a random number generator that churns out an integer from $1$ to $4$ with equal probability, each number corresponding to motion in one of the four directions on a $2D$ plane. [Note: This is far from the most elegant/optimal way to simulate a RW, but it may be the most intuitive/instructive, and that is our goal here.]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x,y=0,0\n",
    "h=1\n",
    "X,Y=[x],[y]\n",
    "n=100\n",
    "\n",
    "for i in range(n):\n",
    "    a=randint(1,5)\n",
    "    if a==1:\n",
    "        x+=h\n",
    "        X.append(x)\n",
    "        Y.append(y)\n",
    "    elif a==2:\n",
    "        x-=h\n",
    "        X.append(x)\n",
    "        Y.append(y)\n",
    "    elif a==3:\n",
    "        y+=h\n",
    "        X.append(x)\n",
    "        Y.append(y)\n",
    "    elif a==4:\n",
    "        y-=h\n",
    "        X.append(x)\n",
    "        Y.append(y)\n",
    "\n",
    "plt.figure(figsize = (8, 8))\n",
    "plt.plot(X, Y, 'bo-', linewidth = 1)\n",
    "plt.plot(0, 0, 'go', ms = 12, label = 'Start')\n",
    "plt.plot(X[-1], Y[-1], 'ro', ms = 12, label = 'End')\n",
    "plt.axis('equal')\n",
    "plt.legend(fontsize=15)\n",
    "plt.title('Figure 1: Simple random walk of length ' + str(n), fontsize=14, fontweight='bold')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## You Can Never Go Back: A Non-Reversing Random Walk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Making things a little more complicated for ourselves, let us now try to model a walk where you can move in any direction you want, except for turning around immediately or taking a step backward: this is called a **non-reversing random walk**. This is slightly more complicated, as at every step, the walker must have a memory of his previous location, like the flip-flop element in electronic circuits. This is unlike the simple random walk model, which is what is called a Markov chain process - every step is independent, and has no connection or causal link with the previous process. The non-reversing walk is not a Markov chain process, it has a memory that goes back one iteration. How to model this?\n",
    "\n",
    "One way is using something called **rejection sampling**. You still go the random walk way, but every time your algorithm generates a movement that is not allowed (i.e. left, when you have just moved right, or up when you have just moved down, etc.), you reject that iteration and run the loop again. In this way, the only movements that show up in your actual working program are the ones that pass this test. However, the problem is attrition: you lose computer power, time and loop iterations doing this, and if you wanted a path of 50 iterations (say), you may not eventually get 50 steps, because of the (illegal!) loops you discarded along the way. Here, we have grouped (up,down) (corresponding to $(1,4)$) and (left,right) (corresponding to $(2,3)$) as complementary moves that are not allowed in successive iterations, as their sums add up to $5$ - whenever this happens, a loop is discarded, and we start again.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x,y=0,0\n",
    "h=1\n",
    "X,Y=[x],[y]\n",
    "n=100\n",
    "steps=0\n",
    "a1=randint(1,5)\n",
    "for i in range(n):\n",
    "    a=randint(1,5)\n",
    "    if a+a1==5: continue\n",
    "    if a==1:\n",
    "        x+=h\n",
    "        X.append(x)\n",
    "        Y.append(y)\n",
    "    elif a==4:\n",
    "        x-=h\n",
    "        X.append(x)\n",
    "        Y.append(y)\n",
    "    elif a==2:\n",
    "        y+=h\n",
    "        X.append(x)\n",
    "        Y.append(y)\n",
    "    elif a==3:\n",
    "        y-=h\n",
    "        X.append(x)\n",
    "        Y.append(y)\n",
    "    a1=a\n",
    "    steps+=1\n",
    "\n",
    "plt.figure(figsize = (8, 8))\n",
    "plt.plot(X, Y, 'bo-', linewidth = 1)\n",
    "plt.plot(0, 0, 'go', ms = 12, label = 'Start')\n",
    "plt.plot(X[-1], Y[-1], 'ro', ms = 12, label = 'End')\n",
    "plt.axis('equal')\n",
    "plt.legend(fontsize=15)\n",
    "plt.title('Figure 2: Non-reversing RW of length ' + str(steps) + ' - attempted length ' + str(n) + ' (rejection sampling)', fontsize=14, fontweight='bold')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A better (and more efficient) way of simulating a non-reversing walk is by restricting the sample space of feasible movements for our walker in the first place - this incorporates the memory of the previous position in a less attritional way. For our next program, before we generate the random number that tells our walker which way to go, we first cross-reference with our memory variable, which stores the last known position, and restrict the space of possible movements, **BEFORE** using the random number generator. Only then do we generate the next step for our walk. This way, we always end up with the path of the length we want.  \n",
    "\n",
    "[Note: One thing to note is the qualitative difference between a simple RW and a non-reversing RW - in the first case, there are many open ends, but in a non-reversing walk, it's all closed loops. We should expect this as the only way an open end is generated is if the walker goes forward and then comes back along the same path and then goes elsewhere - this is specifically prohibited for a non-reversing walk. The only way it can get out of a hole is by creating a closed loop.]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y = 0, 0\n",
    "X, Y = [x], [y]\n",
    "prevpos = [(0,0)]\n",
    "steps=0\n",
    "for i in range(n):\n",
    "    deltas = [(1,0), (0,1), (-1,0), (0,-1)] #these are the increments\n",
    "    deltas_feasible = [] #these store the list with all the feasible increments for step i\n",
    "    for dx, dy in deltas:\n",
    "        if (X[-1] + dx, Y[-1] + dy) not in prevpos:  \n",
    "            deltas_feasible.append((dx,dy))\n",
    "    dx, dy = deltas_feasible[randint(0,len(deltas_feasible))]\n",
    "    X.append(X[-1] + dx)\n",
    "    Y.append(Y[-1] + dy)\n",
    "    prevpos = [(X[-2], Y[-2])]\n",
    "    steps+=1\n",
    "\n",
    "plt.figure(figsize = (8, 8))\n",
    "plt.plot(X, Y, 'bo-', linewidth = 1)\n",
    "plt.plot(0, 0, 'go', ms = 12, label = 'Start')\n",
    "plt.plot(X[-1], Y[-1], 'ro', ms = 12, label = 'End')\n",
    "plt.axis('equal')\n",
    "plt.legend(fontsize=15)\n",
    "plt.title('Figure 3: Non-reversing RW of length ' + str(steps), fontsize=14, fontweight='bold')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Avoiding Oneself: The Rosenbluth Self-Avoiding Walk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We shall now proceed to the meat of our presentation - a **self-avoiding walk**. In simplest terms, it is a random walk that never intersects itself. Our task is the same as that for the famous mobile game *Snake* - the moment the snake bites itself, we are done. [Note: There are more things to be taken into account regarding the ergodicity of our distribution of SAWs and their relative prevalence within that distribution, but we shall come to this later.]\n",
    "\n",
    "The self-avoiding walk is an extremely good model for **linear long-chain polymers in a good solvent** - technically, we can say that the two problems belong in the same **universality class**, as has been rigorously proved by EW Montroll in 1950 and Mark Fisher in 1974 in a modern renormalization group paradigm. As such, figuring out a way to simulate SAWs effectively can help us understand polymer conformations. Considering polymers lie at the heart of almost all of biology and industrial chemistry, this problem is clearly well-motivated. \n",
    "\n",
    "As should be clear here, in this case, our walker needs to remember not just the immediately previous point, but all the points it has visited. A simple algorithm for how to simulate this was given by **Marshall and Arianna Rosenbluth** in 1955, and it goes as follows:\n",
    "\n",
    "> Set the origin as the starting vertex, and recursively choose the next vertex from the set of nearest neighbors that are not occupied by the walk. If there are no nearest neighbors available, then the walk is rejected and a new walk is initiated at the origin.\n",
    "\n",
    "This is similar to what we did for non-reversing random walks, but now we need to store all the previous positions in our memory. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def rosenbluth_saw(n):\n",
    "    x, y = [0], [0]\n",
    "    positions = [(0,0)]  #positions is a list that stores all sites visited by the walk\n",
    "    stuck = 0 #stuck is a Boolean variable that records if we are able to find an SAW of required length, or get stuck\n",
    "    for i in range(n):\n",
    "        deltas = [(1,0), (0,1), (-1,0), (0,-1)]\n",
    "        deltas_feasible = []  #deltas_feasible stores the available directions \n",
    "        for dx, dy in deltas:\n",
    "            if (x[-1] + dx, y[-1] + dy) not in positions:  #checks if direction leads to a site not visited before\n",
    "                deltas_feasible.append((dx,dy))\n",
    "        if deltas_feasible:  #checks if there is a direction available\n",
    "            dx, dy = deltas_feasible[np.random.randint(0,len(deltas_feasible))]  #choose a direction at random among available ones\n",
    "            positions.append((x[-1] + dx, y[-1] + dy))\n",
    "            x.append(x[-1] + dx)\n",
    "            y.append(y[-1] + dy)\n",
    "        else:  #in that case the walk is stuck\n",
    "            stuck = 1\n",
    "            steps = i+1\n",
    "            break  #terminate the walk prematurely\n",
    "        steps = n+1\n",
    "    return x, y, stuck, steps\n",
    "\n",
    "n=80\n",
    "x, y, stuck, steps = rosenbluth_saw(n)\n",
    "plt.figure(figsize = (8, 8))\n",
    "plt.plot(x, y, 'bo-', linewidth = 1)\n",
    "plt.plot(0, 0, 'go', ms = 12, label = 'Start')\n",
    "plt.plot(x[-1], y[-1], 'ro', ms = 12, label = 'End')\n",
    "plt.axis('equal')\n",
    "plt.legend()\n",
    "if stuck:\n",
    "    plt.title('Figure 4: Walk stuck at step ' + str(steps) + ' attempting ' + str(n), fontsize=14, fontweight='bold', y = 1.05)\n",
    "else:\n",
    "    plt.title('Figure 4: SAW of length ' + str(n), fontsize=14, fontweight='bold', y = 1.05)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, this is not a rock-solid approach - there will be many iterations when we are not able to generate a self-avoiding walk of the required length. This is especially a problem in $2D$ and gets compounded exponentially as we go to longer and longer lengths. In fact, if we had done a simple rejection sampling of SAWs from a distribution of RWs, like we did first for the non-reversing walk, we would find that for $n=20$, only about one in $1200$ would be a self-avoiding walk. For $n=50$, that falls to about one in $24,000,000,00$ (240 million). (Thankfully, we didn't take this approach!) Imagine what would happen if we try and simulate SAWs of lengths $n\\approx 3000$, which is the length of a polyethylene chain. But how much better does our Rosenbluth algorithm fare? Let us find out!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The time taken to generate this plot is 109.67506909370422  seconds.\n"
     ]
    }
   ],
   "source": [
    "e0=time.time() #this measures when the process for this cell starts\n",
    "def is_saw(x, y, n):\n",
    "    return n+1 == len(set(zip(x,y)))  #creating a set removes duplicates, so it suffices to check the size of the set\n",
    "def rejection_sampling(n, samples):\n",
    "    accepted = 0\n",
    "    for i in range(samples):\n",
    "        x, y, _, _ = rosenbluth_saw(n) #generates a Rosenbluth SAW\n",
    "        if is_saw(x, y, n): #checks if it is a SAW\n",
    "            accepted += 1\n",
    "    return accepted/samples\n",
    "\n",
    "def plot_acceptance(n, samples):\n",
    "    acceptance = []\n",
    "    n_list = [5, 10, 15, 20, 25, 30, 35, 40, 45]\n",
    "    for i in range(1, n):\n",
    "        acceptance.append(rejection_sampling(i, samples)*100)\n",
    "    plt.figure(figsize = (8, 8))\n",
    "    plt.plot(range(1, n), acceptance, 'o-')\n",
    "    plt.grid(True)\n",
    "    ax = plt.axes()\n",
    "    ax.set_xticks(n_list)\n",
    "    ax.yaxis.set_major_formatter(PercentFormatter())\n",
    "    plt.title('Figure 5: Acceptance rate of the rejection sampling method, for Rosenbluth SAWs of different n', \n",
    "              fontsize=14, fontweight='bold', y = 1.05)\n",
    "    plt.show()\n",
    "\n",
    "plot_acceptance(50,10000)\n",
    "elapsed_time=time.time()-e0\n",
    "print(\"The time taken to generate this plot is\", elapsed_time, \" seconds.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, this graph tells us the Rosenbluth algorithm has pretty good acceptance rates for small $n\\approx 20$ - however, the trend of the graph suggests this will fall away as we go up to 100-300, as it is already at $60$ percent for $n=50$. On running this program for $n=150-300$, the times taken were prohibitively long and computing power required was too high for a low-end laptop - so we have kept it to $n=50$ here, which itself took close to 2 minutes. \n",
    "\n",
    "However, for small $n$, we see that the algorithm will be able to generate an SAW on the vast majority of its iterations. This is in stark contrast to the same program for simple random walks left to generate SAWs by chance, where we shall see the acceptance rate falls away exponentially even for $n\\approx 20$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def simple_RW(n): #generates a random walk by the simple algorithm we used at the beginning\n",
    "    x, y = [0], [0]\n",
    "    deltas = [(1,0), (0,1), (-1,0), (0,-1)] #possible directions for the 2D lattice\n",
    "    for _ in range(n):\n",
    "        dx, dy = deltas[np.random.randint(0,4)]   #direction chosen at random\n",
    "        x.append(x[-1] + dx)\n",
    "        y.append(y[-1] + dy)\n",
    "    return x, y\n",
    "\n",
    "def rejection_sampling_simple(n, samples):\n",
    "    accepted = 0\n",
    "    for i in range(samples):\n",
    "        x, y= simple_RW(n) #generates a Rosenbluth SAW\n",
    "        if is_saw(x, y, n): #checks if it is a SAW\n",
    "            accepted += 1\n",
    "    return accepted/samples\n",
    "\n",
    "def plot_acceptance(n, samples):\n",
    "    acceptance = []\n",
    "    for i in range(1, n):\n",
    "        acceptance.append(rejection_sampling_simple(i, samples)*100)\n",
    "    plt.figure(figsize = (8, 8))\n",
    "    plt.plot(range(1, n), acceptance, 'o-')\n",
    "    plt.grid(True)\n",
    "    ax = plt.axes()\n",
    "    ax.set_xticks(list(range(1,n)))\n",
    "    ax.yaxis.set_major_formatter(PercentFormatter())\n",
    "    plt.title('Figure 6: Acceptance rate of the rejection sampling method, for simple RWs of different n', \n",
    "              fontsize=14, fontweight='bold', y = 1.05)\n",
    "    plt.show()\n",
    "\n",
    "plot_acceptance(20,10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Short Note on Paul Flory, father of polymers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Much of the most influential theoretical work on polymers was done by **Paul Flory**, who used an ingenious combination of dimensional arguments, physical intuition and field theoretic calculations to show that the mean squared end-to-end distance for a polymer (or SAW) varies as\n",
    "\\begin{equation}\n",
    "    R\\propto N^{3/(d+2)},\n",
    "\\end{equation}\n",
    "where $d$ is the naive dimension (NOT the fractal dimension) of the polymer being studied. In our case $d=2$, so we get the relation, \n",
    "\\begin{equation}\n",
    "    R^2\\propto N^{3/2}\n",
    "\\end{equation}\n",
    "As we can see, self-avoiding walks stretch out more than simple random walks (which had a $R^2\\propto N$ relation), on average - this is called the excluded volume effect, and we should expect this! After all, simple random walks can twist over and into themselves and curl up much tighter than a self-avoiding walk, which can never intersect itself - in fact, in some polymers, there is a repulsion effect between non-adjacent monomer units, which further contributes to stretching out. Especially in a good solvent, the monomer units prefer to associate with solvent molecules more than other monomer units, which can further facilitate this stretching out.\n",
    "\n",
    "Flory won a Nobel Prize in 1974 for his work in this field, and his Nobel lecture on **\"Spatial Configuration of Macromolecular Chains\"** is worth reading for those interested in polymer science. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## First Among Equals: A Challenge for Computational Estimation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All well and good. But since we seem to have a fairly decent algorithm to generate SAWs, can we now start to computationally estimate global quantities associated with SAWs, and by extension, polymers? Not so fast. In our discussion, we have conveniently assumed that all SAWs are created equal, whereas in reality, they are not! Different SAWs have different probabilities of occurrence, depending on their geometry. Let us see how.\n",
    "\n",
    "<img src=https://i.imgur.com/p3GqPeL.png width=\"600\">\n",
    "\n",
    "Consider the two 4-step SAWs shown above. What are their probabilities of occurrence?\n",
    "\n",
    "For the first one, when it starts, it has four possible directions to move in - so probability for first step=$\\frac{1}{4}$. The next step can move in any direction but backward, so probability for second step=$\\frac{1}{3}$. Same for third step = $\\frac{1}{3}$. But for fourth step, it only has two directions to go in, left or up, without intersections, so probability =$\\frac{1}{2}$ - so total probability for this walk is $P_1=\\frac{1}{4}\\frac{1}{3}\\frac{1}{3}\\frac{1}{2}$.\n",
    "\n",
    "For the second one, when it starts, as usual, it has four steps to choose from, so probability of first step=$\\frac{1}{4}$. But for all the subsequent steps along the straight line, the walker will always have three options to choose from, i.e. any direction but backward, so probability for 2nd to 4th steps, probability=$\\frac{1}{3}$, so total probability $P_2=\\frac{1}{4}\\frac{1}{3}\\frac{1}{3}\\frac{1}{3}$. \n",
    "\n",
    "As we can see, $P_2>P_1$, in fact the second walk is $1.5$ times as likely to occur if we run a program as the first one. So based on this, we can assign a weight $W_i=1/P_i$ for every SAW, and when we compute any quantity, e.g. $R^2$, we have to make sure to compute a weighted average, or what is called **importance sampling**. \n",
    "\n",
    "In the following program, we shall try to simulate $N=1000$ samples for a range of SAW of lengths going from $n=2,20$, compute the mean squared end-to-end distance, and compare the relative error with the standard values found from the literature. \n",
    "\n",
    "[Note: We have started $n$ from 2 because you need at least 2 points to have 1 step, i.e. a walk.]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def saw_weights(n):\n",
    "    x, y = [0], [0] \n",
    "    positions = [(0,0)]\n",
    "    weights = []\n",
    "    stuck = 0\n",
    "    for i in range(n):\n",
    "        deltas = [(1,0), (0,1), (-1,0), (0,-1)]\n",
    "        deltas_feasible = []\n",
    "        for dx, dy in deltas:\n",
    "            if (x[-1] + dx, y[-1] + dy) not in positions:\n",
    "                deltas_feasible.append((dx,dy))\n",
    "        if deltas_feasible:\n",
    "            weights.append(len(deltas_feasible))  #add inverse probability to weights\n",
    "            dx, dy = deltas_feasible[randint(0,len(deltas_feasible))]\n",
    "            positions.append((x[-1] + dx, y[-1] + dy))\n",
    "            x.append(x[-1] + dx)\n",
    "            y.append(y[-1] + dy)\n",
    "        else:\n",
    "            stuck = 1\n",
    "            break\n",
    "    return x, y, stuck, weights\n",
    "\n",
    "def importance_sampling(n, samples):\n",
    "    saw = 0\n",
    "    weight_list = []  #stores weight of each SAW\n",
    "    norm_list = []  #stores squared end-to-end distance of each SAW\n",
    "    while saw < samples:  \n",
    "        x, y, stuck, weights = saw_weights(n)\n",
    "        weights=np.array(weights)\n",
    "        if not stuck:\n",
    "            weight_list.append(np.prod(weights))  #computes final weight of the walk as product of its weights\n",
    "            norm_list.append(x[-1]**2 + y[-1]**2)  #computes squared end-to-end distance\n",
    "            saw += 1\n",
    "    weight_list = np.array(weight_list)\n",
    "    norm_list = np.array(norm_list)\n",
    "    estim = np.sum(weight_list * norm_list) / np.sum(weight_list)  #computes estimate (vectorized)\n",
    "    std = np.sqrt(np.sum( (weight_list / np.sum(weight_list))**2 * (norm_list - estim)**2 )) #computes std (vectorized)\n",
    "    return (estim, std) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The time taken to generated this plot is 2.517913579940796 seconds\n"
     ]
    }
   ],
   "source": [
    "e0=time.time()\n",
    "sq_dis = [2.66667,4.55556,7.04,9.56338,12.5744,15.5562,19.0128,22.4114,26.2425,30.0177,34.187,38.3043,42.7864,\n",
    "          47.2177,51.9925,56.7164,61.7665,66.7658,72.0765] # hardcoded list of exact mean-squared distances obtained from the literature\n",
    "sq_dis=np.array(sq_dis)\n",
    "rsq = []\n",
    "relative_error= []\n",
    "for i in range(19):\n",
    "    estim, _ = importance_sampling(i+2, 1000)\n",
    "    rsq.append(estim)\n",
    "    relative_error.append(abs(estim-sq_dis[i])/sq_dis[i]*100)\n",
    "plt.figure(figsize = (8, 8))\n",
    "plt.plot(range(2, 21), relative_error, 'o-')\n",
    "plt.grid(True)\n",
    "ax = plt.axes()\n",
    "ax.set_xticks(list(range(2,21)))\n",
    "ax.yaxis.set_major_formatter(PercentFormatter())\n",
    "plt.title('Figure 7: Relative errors of the estimate of the squared end-to-end distance, for different n', \n",
    "              fontsize=14, fontweight='bold', y = 1.05)\n",
    "plt.show()\n",
    "elapsed_time=time.time()-e0\n",
    "print(\"The time taken to generated this plot is\", elapsed_time, \"seconds\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is not at all what we should expect. For $n>10$, we see a sharp spike in the relative error - it has been suggested in the literature that this kind of instability exists in the Rosenbluth algorithm, and that we may need significantly greater processing power to be able to generate enough samples for this to be a useful computational estimate. For $n=10$, there are about the order of $10^7$ unique walks - for $n=11$, this jumps up to $10^8$, and maybe $N=10^3$ may no longer be good enough to get an accurate idea of estimates over the whole sample space. However, at least we see that upto $n=10$, this gives us good correlation with the expected mean squared end-to-end distances. \n",
    "\n",
    "To end this exploration, we end with two plots, one for the walks we generated from $n=(2,10)$ and one for the curve $R=N^{3/2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x206e19447c0>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(2,9,num=8)\n",
    "plt.plot(x, rsq[1:9])\n",
    "plt.plot(x, x**1.5)\n",
    "plt.legend([\"Computed Estimate of R^2\", \"R^2=N^(3/2)\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "/* Put everything inside the global mpl namespace */\n",
       "/* global mpl */\n",
       "window.mpl = {};\n",
       "\n",
       "mpl.get_websocket_type = function () {\n",
       "    if (typeof WebSocket !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof MozWebSocket !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert(\n",
       "            'Your browser does not have WebSocket support. ' +\n",
       "                'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "                'Firefox 4 and 5 are also supported but you ' +\n",
       "                'have to enable WebSockets in about:config.'\n",
       "        );\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure = function (figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = this.ws.binaryType !== undefined;\n",
       "\n",
       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById('mpl-warnings');\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent =\n",
       "                'This browser does not support binary websocket messages. ' +\n",
       "                'Performance may be slow.';\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = document.createElement('div');\n",
       "    this.root.setAttribute('style', 'display: inline-block');\n",
       "    this._root_extra_style(this.root);\n",
       "\n",
       "    parent_element.appendChild(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen = function () {\n",
       "        fig.send_message('supports_binary', { value: fig.supports_binary });\n",
       "        fig.send_message('send_image_mode', {});\n",
       "        if (fig.ratio !== 1) {\n",
       "            fig.send_message('set_dpi_ratio', { dpi_ratio: fig.ratio });\n",
       "        }\n",
       "        fig.send_message('refresh', {});\n",
       "    };\n",
       "\n",
       "    this.imageObj.onload = function () {\n",
       "        if (fig.image_mode === 'full') {\n",
       "            // Full images could contain transparency (where diff images\n",
       "            // almost always do), so we need to clear the canvas so that\n",
       "            // there is no ghosting.\n",
       "            fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n",
       "        }\n",
       "        fig.context.drawImage(fig.imageObj, 0, 0);\n",
       "    };\n",
       "\n",
       "    this.imageObj.onunload = function () {\n",
       "        fig.ws.close();\n",
       "    };\n",
       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._init_header = function () {\n",
       "    var titlebar = document.createElement('div');\n",
       "    titlebar.classList =\n",
       "        'ui-dialog-titlebar ui-widget-header ui-corner-all ui-helper-clearfix';\n",
       "    var titletext = document.createElement('div');\n",
       "    titletext.classList = 'ui-dialog-title';\n",
       "    titletext.setAttribute(\n",
       "        'style',\n",
       "        'width: 100%; text-align: center; padding: 3px;'\n",
       "    );\n",
       "    titlebar.appendChild(titletext);\n",
       "    this.root.appendChild(titlebar);\n",
       "    this.header = titletext;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function (_canvas_div) {};\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function (_canvas_div) {};\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function () {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = (this.canvas_div = document.createElement('div'));\n",
       "    canvas_div.setAttribute(\n",
       "        'style',\n",
       "        'border: 1px solid #ddd;' +\n",
       "            'box-sizing: content-box;' +\n",
       "            'clear: both;' +\n",
       "            'min-height: 1px;' +\n",
       "            'min-width: 1px;' +\n",
       "            'outline: 0;' +\n",
       "            'overflow: hidden;' +\n",
       "            'position: relative;' +\n",
       "            'resize: both;'\n",
       "    );\n",
       "\n",
       "    function on_keyboard_event_closure(name) {\n",
       "        return function (event) {\n",
       "            return fig.key_event(event, name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    canvas_div.addEventListener(\n",
       "        'keydown',\n",
       "        on_keyboard_event_closure('key_press')\n",
       "    );\n",
       "    canvas_div.addEventListener(\n",
       "        'keyup',\n",
       "        on_keyboard_event_closure('key_release')\n",
       "    );\n",
       "\n",
       "    this._canvas_extra_style(canvas_div);\n",
       "    this.root.appendChild(canvas_div);\n",
       "\n",
       "    var canvas = (this.canvas = document.createElement('canvas'));\n",
       "    canvas.classList.add('mpl-canvas');\n",
       "    canvas.setAttribute('style', 'box-sizing: content-box;');\n",
       "\n",
       "    this.context = canvas.getContext('2d');\n",
       "\n",
       "    var backingStore =\n",
       "        this.context.backingStorePixelRatio ||\n",
       "        this.context.webkitBackingStorePixelRatio ||\n",
       "        this.context.mozBackingStorePixelRatio ||\n",
       "        this.context.msBackingStorePixelRatio ||\n",
       "        this.context.oBackingStorePixelRatio ||\n",
       "        this.context.backingStorePixelRatio ||\n",
       "        1;\n",
       "\n",
       "    this.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "    if (this.ratio !== 1) {\n",
       "        fig.send_message('set_dpi_ratio', { dpi_ratio: this.ratio });\n",
       "    }\n",
       "\n",
       "    var rubberband_canvas = (this.rubberband_canvas = document.createElement(\n",
       "        'canvas'\n",
       "    ));\n",
       "    rubberband_canvas.setAttribute(\n",
       "        'style',\n",
       "        'box-sizing: content-box; position: absolute; left: 0; top: 0; z-index: 1;'\n",
       "    );\n",
       "\n",
       "    var resizeObserver = new ResizeObserver(function (entries) {\n",
       "        var nentries = entries.length;\n",
       "        for (var i = 0; i < nentries; i++) {\n",
       "            var entry = entries[i];\n",
       "            var width, height;\n",
       "            if (entry.contentBoxSize) {\n",
       "                if (entry.contentBoxSize instanceof Array) {\n",
       "                    // Chrome 84 implements new version of spec.\n",
       "                    width = entry.contentBoxSize[0].inlineSize;\n",
       "                    height = entry.contentBoxSize[0].blockSize;\n",
       "                } else {\n",
       "                    // Firefox implements old version of spec.\n",
       "                    width = entry.contentBoxSize.inlineSize;\n",
       "                    height = entry.contentBoxSize.blockSize;\n",
       "                }\n",
       "            } else {\n",
       "                // Chrome <84 implements even older version of spec.\n",
       "                width = entry.contentRect.width;\n",
       "                height = entry.contentRect.height;\n",
       "            }\n",
       "\n",
       "            // Keep the size of the canvas and rubber band canvas in sync with\n",
       "            // the canvas container.\n",
       "            if (entry.devicePixelContentBoxSize) {\n",
       "                // Chrome 84 implements new version of spec.\n",
       "                canvas.setAttribute(\n",
       "                    'width',\n",
       "                    entry.devicePixelContentBoxSize[0].inlineSize\n",
       "                );\n",
       "                canvas.setAttribute(\n",
       "                    'height',\n",
       "                    entry.devicePixelContentBoxSize[0].blockSize\n",
       "                );\n",
       "            } else {\n",
       "                canvas.setAttribute('width', width * fig.ratio);\n",
       "                canvas.setAttribute('height', height * fig.ratio);\n",
       "            }\n",
       "            canvas.setAttribute(\n",
       "                'style',\n",
       "                'width: ' + width + 'px; height: ' + height + 'px;'\n",
       "            );\n",
       "\n",
       "            rubberband_canvas.setAttribute('width', width);\n",
       "            rubberband_canvas.setAttribute('height', height);\n",
       "\n",
       "            // And update the size in Python. We ignore the initial 0/0 size\n",
       "            // that occurs as the element is placed into the DOM, which should\n",
       "            // otherwise not happen due to the minimum size styling.\n",
       "            if (width != 0 && height != 0) {\n",
       "                fig.request_resize(width, height);\n",
       "            }\n",
       "        }\n",
       "    });\n",
       "    resizeObserver.observe(canvas_div);\n",
       "\n",
       "    function on_mouse_event_closure(name) {\n",
       "        return function (event) {\n",
       "            return fig.mouse_event(event, name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mousedown',\n",
       "        on_mouse_event_closure('button_press')\n",
       "    );\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mouseup',\n",
       "        on_mouse_event_closure('button_release')\n",
       "    );\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mousemove',\n",
       "        on_mouse_event_closure('motion_notify')\n",
       "    );\n",
       "\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mouseenter',\n",
       "        on_mouse_event_closure('figure_enter')\n",
       "    );\n",
       "    rubberband_canvas.addEventListener(\n",
       "        'mouseleave',\n",
       "        on_mouse_event_closure('figure_leave')\n",
       "    );\n",
       "\n",
       "    canvas_div.addEventListener('wheel', function (event) {\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        on_mouse_event_closure('scroll')(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.appendChild(canvas);\n",
       "    canvas_div.appendChild(rubberband_canvas);\n",
       "\n",
       "    this.rubberband_context = rubberband_canvas.getContext('2d');\n",
       "    this.rubberband_context.strokeStyle = '#000000';\n",
       "\n",
       "    this._resize_canvas = function (width, height, forward) {\n",
       "        if (forward) {\n",
       "            canvas_div.style.width = width + 'px';\n",
       "            canvas_div.style.height = height + 'px';\n",
       "        }\n",
       "    };\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    this.rubberband_canvas.addEventListener('contextmenu', function (_e) {\n",
       "        event.preventDefault();\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus() {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function () {\n",
       "    var fig = this;\n",
       "\n",
       "    var toolbar = document.createElement('div');\n",
       "    toolbar.classList = 'mpl-toolbar';\n",
       "    this.root.appendChild(toolbar);\n",
       "\n",
       "    function on_click_closure(name) {\n",
       "        return function (_event) {\n",
       "            return fig.toolbar_button_onclick(name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    function on_mouseover_closure(tooltip) {\n",
       "        return function (event) {\n",
       "            if (!event.currentTarget.disabled) {\n",
       "                return fig.toolbar_button_onmouseover(tooltip);\n",
       "            }\n",
       "        };\n",
       "    }\n",
       "\n",
       "    fig.buttons = {};\n",
       "    var buttonGroup = document.createElement('div');\n",
       "    buttonGroup.classList = 'mpl-button-group';\n",
       "    for (var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            /* Instead of a spacer, we start a new button group. */\n",
       "            if (buttonGroup.hasChildNodes()) {\n",
       "                toolbar.appendChild(buttonGroup);\n",
       "            }\n",
       "            buttonGroup = document.createElement('div');\n",
       "            buttonGroup.classList = 'mpl-button-group';\n",
       "            continue;\n",
       "        }\n",
       "\n",
       "        var button = (fig.buttons[name] = document.createElement('button'));\n",
       "        button.classList = 'mpl-widget';\n",
       "        button.setAttribute('role', 'button');\n",
       "        button.setAttribute('aria-disabled', 'false');\n",
       "        button.addEventListener('click', on_click_closure(method_name));\n",
       "        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n",
       "\n",
       "        var icon_img = document.createElement('img');\n",
       "        icon_img.src = '_images/' + image + '.png';\n",
       "        icon_img.srcset = '_images/' + image + '_large.png 2x';\n",
       "        icon_img.alt = tooltip;\n",
       "        button.appendChild(icon_img);\n",
       "\n",
       "        buttonGroup.appendChild(button);\n",
       "    }\n",
       "\n",
       "    if (buttonGroup.hasChildNodes()) {\n",
       "        toolbar.appendChild(buttonGroup);\n",
       "    }\n",
       "\n",
       "    var fmt_picker = document.createElement('select');\n",
       "    fmt_picker.classList = 'mpl-widget';\n",
       "    toolbar.appendChild(fmt_picker);\n",
       "    this.format_dropdown = fmt_picker;\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = document.createElement('option');\n",
       "        option.selected = fmt === mpl.default_extension;\n",
       "        option.innerHTML = fmt;\n",
       "        fmt_picker.appendChild(option);\n",
       "    }\n",
       "\n",
       "    var status_bar = document.createElement('span');\n",
       "    status_bar.classList = 'mpl-message';\n",
       "    toolbar.appendChild(status_bar);\n",
       "    this.message = status_bar;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', { width: x_pixels, height: y_pixels });\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.send_message = function (type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function () {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_save = function (fig, _msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function (fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1], msg['forward']);\n",
       "        fig.send_message('refresh', {});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function (fig, msg) {\n",
       "    var x0 = msg['x0'] / fig.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;\n",
       "    var x1 = msg['x1'] / fig.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0,\n",
       "        0,\n",
       "        fig.canvas.width / fig.ratio,\n",
       "        fig.canvas.height / fig.ratio\n",
       "    );\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function (fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function (fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch (cursor) {\n",
       "        case 0:\n",
       "            cursor = 'pointer';\n",
       "            break;\n",
       "        case 1:\n",
       "            cursor = 'default';\n",
       "            break;\n",
       "        case 2:\n",
       "            cursor = 'crosshair';\n",
       "            break;\n",
       "        case 3:\n",
       "            cursor = 'move';\n",
       "            break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_message = function (fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function (fig, _msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function (fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n",
       "    for (var key in msg) {\n",
       "        if (!(key in fig.buttons)) {\n",
       "            continue;\n",
       "        }\n",
       "        fig.buttons[key].disabled = !msg[key];\n",
       "        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n",
       "    if (msg['mode'] === 'PAN') {\n",
       "        fig.buttons['Pan'].classList.add('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    } else if (msg['mode'] === 'ZOOM') {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.add('active');\n",
       "    } else {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message('ack', {});\n",
       "};\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function (fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = 'image/png';\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src\n",
       "                );\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data\n",
       "            );\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        } else if (\n",
       "            typeof evt.data === 'string' &&\n",
       "            evt.data.slice(0, 21) === 'data:image/png;base64'\n",
       "        ) {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig['handle_' + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\n",
       "                \"No handler for the '\" + msg_type + \"' message type: \",\n",
       "                msg\n",
       "            );\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\n",
       "                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n",
       "                    e,\n",
       "                    e.stack,\n",
       "                    msg\n",
       "                );\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "};\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function (e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e) {\n",
       "        e = window.event;\n",
       "    }\n",
       "    if (e.target) {\n",
       "        targ = e.target;\n",
       "    } else if (e.srcElement) {\n",
       "        targ = e.srcElement;\n",
       "    }\n",
       "    if (targ.nodeType === 3) {\n",
       "        // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "    }\n",
       "\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    var boundingRect = targ.getBoundingClientRect();\n",
       "    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n",
       "    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n",
       "\n",
       "    return { x: x, y: y };\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys(original) {\n",
       "    return Object.keys(original).reduce(function (obj, key) {\n",
       "        if (typeof original[key] !== 'object') {\n",
       "            obj[key] = original[key];\n",
       "        }\n",
       "        return obj;\n",
       "    }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function (event, name) {\n",
       "    var canvas_pos = mpl.findpos(event);\n",
       "\n",
       "    if (name === 'button_press') {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * this.ratio;\n",
       "    var y = canvas_pos.y * this.ratio;\n",
       "\n",
       "    this.send_message(name, {\n",
       "        x: x,\n",
       "        y: y,\n",
       "        button: event.button,\n",
       "        step: event.step,\n",
       "        guiEvent: simpleKeys(event),\n",
       "    });\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function (_event, _name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.key_event = function (event, name) {\n",
       "    // Prevent repeat events\n",
       "    if (name === 'key_press') {\n",
       "        if (event.which === this._key) {\n",
       "            return;\n",
       "        } else {\n",
       "            this._key = event.which;\n",
       "        }\n",
       "    }\n",
       "    if (name === 'key_release') {\n",
       "        this._key = null;\n",
       "    }\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which !== 17) {\n",
       "        value += 'ctrl+';\n",
       "    }\n",
       "    if (event.altKey && event.which !== 18) {\n",
       "        value += 'alt+';\n",
       "    }\n",
       "    if (event.shiftKey && event.which !== 16) {\n",
       "        value += 'shift+';\n",
       "    }\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n",
       "    return false;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function (name) {\n",
       "    if (name === 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message('toolbar_button', { name: name });\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";/* global mpl */\n",
       "\n",
       "var comm_websocket_adapter = function (comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function () {\n",
       "        comm.close();\n",
       "    };\n",
       "    ws.send = function (m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function (msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data']);\n",
       "    });\n",
       "    return ws;\n",
       "};\n",
       "\n",
       "mpl.mpl_figure_comm = function (comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = document.getElementById(id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm);\n",
       "\n",
       "    function ondownload(figure, _format) {\n",
       "        window.open(figure.canvas.toDataURL());\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element;\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error('Failed to find cell for figure', id, fig);\n",
       "        return;\n",
       "    }\n",
       "    fig.cell_info[0].output_area.element.one(\n",
       "        'cleared',\n",
       "        { fig: fig },\n",
       "        fig._remove_fig_handler\n",
       "    );\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function (fig, msg) {\n",
       "    var width = fig.canvas.width / fig.ratio;\n",
       "    fig.cell_info[0].output_area.element.off(\n",
       "        'cleared',\n",
       "        fig._remove_fig_handler\n",
       "    );\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable();\n",
       "    fig.parent_element.innerHTML =\n",
       "        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "    fig.close_ws(fig, msg);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.close_ws = function (fig, msg) {\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function (_remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width / this.ratio;\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] =\n",
       "        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message('ack', {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () {\n",
       "        fig.push_to_output();\n",
       "    }, 1000);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function () {\n",
       "    var fig = this;\n",
       "\n",
       "    var toolbar = document.createElement('div');\n",
       "    toolbar.classList = 'btn-toolbar';\n",
       "    this.root.appendChild(toolbar);\n",
       "\n",
       "    function on_click_closure(name) {\n",
       "        return function (_event) {\n",
       "            return fig.toolbar_button_onclick(name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    function on_mouseover_closure(tooltip) {\n",
       "        return function (event) {\n",
       "            if (!event.currentTarget.disabled) {\n",
       "                return fig.toolbar_button_onmouseover(tooltip);\n",
       "            }\n",
       "        };\n",
       "    }\n",
       "\n",
       "    fig.buttons = {};\n",
       "    var buttonGroup = document.createElement('div');\n",
       "    buttonGroup.classList = 'btn-group';\n",
       "    var button;\n",
       "    for (var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            /* Instead of a spacer, we start a new button group. */\n",
       "            if (buttonGroup.hasChildNodes()) {\n",
       "                toolbar.appendChild(buttonGroup);\n",
       "            }\n",
       "            buttonGroup = document.createElement('div');\n",
       "            buttonGroup.classList = 'btn-group';\n",
       "            continue;\n",
       "        }\n",
       "\n",
       "        button = fig.buttons[name] = document.createElement('button');\n",
       "        button.classList = 'btn btn-default';\n",
       "        button.href = '#';\n",
       "        button.title = name;\n",
       "        button.innerHTML = '<i class=\"fa ' + image + ' fa-lg\"></i>';\n",
       "        button.addEventListener('click', on_click_closure(method_name));\n",
       "        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n",
       "        buttonGroup.appendChild(button);\n",
       "    }\n",
       "\n",
       "    if (buttonGroup.hasChildNodes()) {\n",
       "        toolbar.appendChild(buttonGroup);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = document.createElement('span');\n",
       "    status_bar.classList = 'mpl-message pull-right';\n",
       "    toolbar.appendChild(status_bar);\n",
       "    this.message = status_bar;\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = document.createElement('div');\n",
       "    buttongrp.classList = 'btn-group inline pull-right';\n",
       "    button = document.createElement('button');\n",
       "    button.classList = 'btn btn-mini btn-primary';\n",
       "    button.href = '#';\n",
       "    button.title = 'Stop Interaction';\n",
       "    button.innerHTML = '<i class=\"fa fa-power-off icon-remove icon-large\"></i>';\n",
       "    button.addEventListener('click', function (_evt) {\n",
       "        fig.handle_close(fig, {});\n",
       "    });\n",
       "    button.addEventListener(\n",
       "        'mouseover',\n",
       "        on_mouseover_closure('Stop Interaction')\n",
       "    );\n",
       "    buttongrp.appendChild(button);\n",
       "    var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n",
       "    titlebar.insertBefore(buttongrp, titlebar.firstChild);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._remove_fig_handler = function (event) {\n",
       "    var fig = event.data.fig;\n",
       "    fig.close_ws(fig, {});\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function (el) {\n",
       "    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function (el) {\n",
       "    // this is important to make the div 'focusable\n",
       "    el.setAttribute('tabindex', 0);\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    } else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function (event, _name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager) {\n",
       "        manager = IPython.keyboard_manager;\n",
       "    }\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which === 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_save = function (fig, _msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "};\n",
       "\n",
       "mpl.find_output_cell = function (html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i = 0; i < ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code') {\n",
       "            for (var j = 0; j < cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] === html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "};\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel !== null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target(\n",
       "        'matplotlib',\n",
       "        mpl.mpl_figure_comm\n",
       "    );\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Animation\n",
    "\n",
    "%matplotlib notebook\n",
    "import celluloid\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.pylab as pl\n",
    "import numpy as np\n",
    "import numba\n",
    "from numba import jit, njit\n",
    "from celluloid import Camera\n",
    "import ffmpeg\n",
    "from numpy.random import randint, random, choice\n",
    "from IPython.display import HTML\n",
    "from matplotlib.colors import LogNorm\n",
    "from matplotlib import cm\n",
    "\n",
    "np.random.seed(0)\n",
    "n=100\n",
    "fig=plt.figure(figsize=(8, 8))\n",
    "camera = Camera(fig)\n",
    "\n",
    "x, y = [0], [0]\n",
    "positions = [(0,0)]  #positions is a list that stores all sites visited by the walk\n",
    "for i in range(n):\n",
    "    deltas = [(1,0), (0,1), (-1,0), (0,-1)]\n",
    "    deltas_feasible = []  #deltas_feasible stores the available directions \n",
    "    for dx, dy in deltas:\n",
    "        if (x[-1] + dx, y[-1] + dy) not in positions:  #checks if direction leads to a site not visited before\n",
    "            deltas_feasible.append((dx,dy))\n",
    "    if deltas_feasible:  #checks if there is a direction available\n",
    "        dx, dy = deltas_feasible[randint(0,len(deltas_feasible))]  #choose a direction at random among available ones\n",
    "        positions.append((x[-1] + dx, y[-1] + dy))\n",
    "        x.append(x[-1] + dx)\n",
    "        y.append(y[-1] + dy)\n",
    "    else:  #in that case the walk is stuck\n",
    "        break  #terminate the walk prematurely\n",
    "    plt.plot(x, y, 'bo-', linewidth = 1)\n",
    "    plt.plot(0, 0, 'go', ms = 12)\n",
    "    plt.plot(x[-1], y[-1], 'ro', ms = 12)\n",
    "    plt.xlim(min(x)-1, max(x)+1)\n",
    "    plt.ylim(min(y)-1, max(y)+1)\n",
    "    plt.axis('equal')\n",
    "    camera.snap()\n",
    "\n",
    "animation = camera.animate()\n",
    "animation.save('SAW_anim.mp4')\n",
    "HTML(animation.to_html5_video())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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